Senior Math Circles: Geometry III

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1 University of Waterloo Faculty of Mathematics entre for Education in Mathematics and omputing Senior Math ircles: Geometry III eview of Important Facts bout Trigonometry Most famous trig identity: sin 2 x + cos 2 x = 1 sin(x + y) = sin x cos y + cos x sin y sin(x y) = sin x cos y cos x sin y cos(x + y) = cos x cos y sin x sin y cos(x y) = cos x cos y + sin x sin y tan(x + y) = tan(x y) = tan x + tan y 1 tan x tan y tan x tan y 1 + tan x tan y sin(2x) = 2 sin x cos x cos(2x) = cos 2 x sin 2 x tan(2x) = 2 tan x 1 tan 2 x sin( x) = sin x cos( x) = cos x tan( x) = tan x racticing with these facts 1. alculate sin(105 ). 2. alculate cos(22.5 ). 3. Simplify cos(90 + x). 4. Simplify sin(180 x). 5. Simplify cos(90 x).

2 University of Waterloo Faculty of Mathematics entre for Education in Mathematics and omputing roblem Set 2 Senior Math ircles: Geometry III 1. If 0 < x < 90 and tan(2x) = 24, determine the value of sin x. 7 (Source: 2002 Descartes ontest) 2. Determine the values of x, 0 < x < π, for which (Source: 2001 Descartes ontest) 3. (a) rove that sin 2 = 2 tan 1 + tan 2, where 0 < < π. 2 (b) If sin 2 = 4, find tan. 5 (Source: 1998 Descartes ontest) cos 2 x = For what values of θ, π 2 θ π 2, does the equation x2 + (2 sin θ)x + cos 2θ = 0 have real roots? (Source: 1996 Descartes ontest) 5. (a) If tan x = a, 0 < x < π, find sin 2x in terms of a. 2 (b) Given the equations cos x + cos y = 2m (cos x)(cos y) = 3m 2 where m, find an expression for sin 2 x + sin 2 y in terms of m. (Source: 2000 Descartes ontest) 6. Determine the points of intersection of the curves defined by y = 8 cos x + 5 tan x and y = 5 sec x + 2 sin 2x for 0 x 2π. (Source: 1997 Descartes ontest) 7. rove that there are no real values of x such that 2 sin x = x 2 4x + 6. (Source: 1993 Euclid ontest) 8. In the quadrilateral D, angles D and D are right angles. lso, D = D = 60 and D = 4. Determine which is greater: D + or D +. (Source: 1997 Descartes ontest) D

3 9. has = 30, = 150, and = Determine the length of. (Source: 1976 Euclid ontest) 10. (a) The quadrilateral D has = 5, = 6 2 and D = 7. If D = 8 and = 105, determine the length of D. D (b) rove the identity: sin( ) sin( + ) = sin 2 sin 2. (Source: 1999 Descartes ontest) 11. In, a = 3 2, b = 4 2, = 45, and is obtuse. Determine c. (Source: 1977 Euclid ontest) 12. (a) Determine the constants a and b so that cos2 θ is equal to a + b sin θ for all values 1 2 sin θ of θ. (b) Find all values of x, 0 x 2π, which satisfy sin 2 x + cos x + 1 = 0. (Source: 1977 Euclid ontest) 13. Find all angles x in the interval π x π such that sin 2x + sin 3x = sin x. 14. In, sin = 3 and sin = 1. What is the ratio :? 5 4 (Source: 1978 Euclid ontest) 15. Two ships, and leave port at the same time, travelling at constant speeds of 20 km/h and 32 km/h, respectively. If the angle separating their paths is 60, what is the distance between their positions after 2.5 hours? (Source: 1998 Descartes ontest) has its vertices on a circle with radius 2 and its centre at O, as shown. If = 3, calculate the cosine of. (Source: 2000 Descartes ontest) O 17. In triangle, given that b cos = a cos, prove that a = b. (Source: 1994 Euclid ontest)

4 18. The length of the hands of a clock are 6 cm and 4 cm, respectively. What is the distance between the tips of the hands at two o clock? (Source: 1978 Euclid ontest) 19. Starting at 10:00 a.m. from, a runner runs due north at a speed of 10 km/h. Starting 30 minutes later from, which is 25 km east of, a cyclist travels northwest at a constant speed. The runner and the cyclist arrive at the same point at the same time. Determine the speed of the bicycle. (Source: 1996 Euclid ontest) In, angle is twice angle. rove that a 2 = b(b + c). 21. Let a be the length of a side and b the length of a diagonal in the regular pentagon ST. rove that b a a b = 1. (Source: 1998 Descartes ontest) a b T S 22. regular octagon, DEF GH, is inscribed in a circle of radius 1. rove that the product of the lengths of the line segments joining to each of the other vertices equals 8. (Source: 1979 Euclid ontest) 23. n equilateral triangle of side length 1 is inscribed in the rectangle so that lies on and lies on, as shown. rove that the area of is equal to the sum of the areas of and. (Source: 2002 Descartes ontest) 24. Two circles of radii 4 and 2 have their centres 4 units apart and intersect at X and Y. line drawn through X cuts the circles at and, and X meets the line of centres produced at an angle of 30. alculate the length of X. (Source: 1977 Euclid ontest) 25. In the diagram, the circles with centres and are tangent externally at T. is a common tangent line. The line of centres also intersects the circles at and S, as shown. and S, when produced, meet at X. rove that XS is a right angle. (Source: 1978 Euclid ontest) T S

5 26. and are tangents to the given circle. If arc S = 4 arc T, what is in degrees? (Source: 1977 Euclid ontest) S T 27. In the diagram, a box in the shape of a cube with side 1.2 m is sitting on a hand cart 1 m from the front end,, of the cart platform. The platform is parallel to the floor. The wheels on the cart have radii 0.4 m. The back end of the platform is lifted so that the point is rotated 15 about the point. What is the minimum height of a doorway through which the cart can pass? (Source: 1996 Euclid ontest) 28. For the equilateral triangle, side is the diameter of a semicircle. oints and create equal arcs = = on the semicircle. Show that line segments and trisect side. (Source: 1995 Euclid ontest) 29. In, = If = 1, = r, and = r 2, where r > 1, prove that r < 2. (Source: 1996 Euclid ontest) 30. In, angle is acute and is any point on. The reflection of point in is point T, and the reflection of point in is point S. Determine the position of point so that the area of T S is a minimum. (Source: 1992 Euclid ontest) 31. The hypotenuse is right triangle is divided into three equal parts by lines and. If = 9 and = 7, what is the length of?

6 For the remaining problems, let, and be the angles of a triangle. 32. rove that 4 cos 2 cos 2 cos rove that 4 sin 2 sin 2 sin 2 = sin + sin + sin. = cos + cos + cos rove that cos 2 + cos 2 + cos cos cos cos = rove that tan + tan + tan = tan tan tan. 36. rove that (1 + tan )(1 + tan ) = 2 if and only if + = π rove that sin 2 + sin 2 + sin 2 = 4 sin sin sin. 38. rove that tan tan + tan tan + tan tan =